There are five positive observations. The average of the first three observations is $8$ and that of the last three observations is $6$. If the average of all five observations is $6$, what is the third observation?

Let the five positive observations be $x_1,x_2,x_3,x_4,\text{ and }{x}_5$.
Given:

1. Average of all $5$ observations is $6$:

$$\frac{x_1+x_2+x_3+x_4+x_5}{5}=6⟹x_1+x_2+x_3+x_4+x_5=30$$

2. Average of the first $3$ observations is $8$:

$$\frac{x_1+x_2+x_3}{3}=8⟹x_1+x_2+x_3=24$$

3. Average of the last $3$ observations is $6$:

$$\frac{x_3+x_4+x_5}{3}=6⟹x_3+x_4+x_5=18$$

Add equations (2) and (3):

$$(x_1+x_2+x_3)+(x_3+x_4+x_5)=24+18=42$$

$$x_1+x_2+x_3+x_4+x_5+x_3=42$$

Substitute the value of the total sum from equation (1) into this expression:

$$30+x_3=42⟹x_3=42-30=12$$

Thus, the third observation is $12$.